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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011 923
Closed-Form Design Formulas for the Equivalent
Circuit Characterization of Ferrite Inductors
Krishna Naishadham
Abstract—Miniature inductors, consisting of thin-wire
solenoidal or toroidal coils wound on a high-permeability soft
ferrite core, find wide application in the filtering of noise or
electromagnetic interference at RF. An understanding of the
high-frequency parasitic and packaging effects of the inductor,
such as stray capacitance, magnetic losses, self-resonance, etc.,
can be gained from an equivalent circuit characterization of
the inductor. In this paper, we present a rigorous experimental
method to fully characterize the RF behavior of ferrite inductors.
The equivalent circuit parameters of the inductor, namely, series
inductance, loss resistance, and stray capacitance, as well as the
effective permeability of the core, are extracted in closed form
from an accurate measurement of the RF impedance, without
recourse to cumbersome optimization procedures usually followed
in equivalent circuit extraction from measured data. We derive
design equations for the equivalent circuit elements based on
the proposed measurement-based model, and rigorously validate
the model using inductors wound on commercial toroidal and
rod-type Ni–Zn ferrite cores.
Index Terms—Electromagnetic compatibility (EMC), electro-
magnetic interference (EMI), equivalent circuit analysis, ferrite
inductors, magnetic materials, permeability, RF inductors.
I. INTRODUCTION
FERRITE inductors, consisting of an insulated copper wire
wound on slug-type, toroidal, or brick-type magnetic cores,
find useful application in filtering noise resulting from electro-
magnetic interference (EMI) at RF. Besides the insertion loss of
the inductor, parasitic effects, such as interturn and interwind-
ing capacitance, self-resonance, dielectric and magnetic losses
of the core, play an important role in the design of such filters.
The distributed capacitance across the coil acts like a lumped
capacitor in shunt with inductor, and results in occurrence of
self-resonance at some frequency, above which the impedance
becomes predominantly capacitive [1], [2]. The useful operat-
ing frequency range of the inductor is thus limited by its self-
resonant frequency (SRF). In automotive electronics, broadband
wireless communications and cable television systems, it is now
required to design inductors with resonant frequency extending
into the RF regime (100 MHz to 1 GHz). Thus, there is a need to
understand the full circuit characterization of the RF inductor,
Manuscript received July 8, 2009; revised February 25, 2010, July 26, 2010,
and January 11, 2011; accepted January 15, 2011. Date of publication May 27,
2011; date of current version November 18, 2011.
The author is with the Sensors and Electromagnetic Applications Labo-
ratory, Georgia Institute of Technology, Atlanta, GA 30080 USA (e-mail:
krishna.naishadham@gtri.gatech.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEMC.2011.2116795
including all the frequency-dependent circuit-parasitic, mate-
rial, and packaging effects.
At RF, direct measurement of the inductance, resistance, and
stray capacitance of the inductor, as functions of frequency,
is difficult. It thus becomes important to develop a technique
for extracting these circuit parameters from measurable physi-
cal parameters, such as insertion loss, resonant frequency, and
impedance. Prior investigations on equivalent circuit charac-
terization of inductors have been limited to frequencies much
lower than the self-resonance, where the circuit parameters can
be assumed as constant. An electromagnetic formulation based
on quasi-static Maxwell’s equations [3] is adequate in this case
to describe the inductor. At high frequencies, the assumptions
of quasi-static analysis are violated. Bartoli et al. [4] proposed
a method for estimating the self-capacitance of an inductor by
measuring its resonant frequency with the assumption that the
inductance near resonance is unchanged from its low-frequency
value. In ferrite inductors, this assumption is valid only if the
SRF is smaller than the intrinsic “rolloff” frequency at which
the real part of the permeability begins to roll off from a constant
low-frequency value, and where the imaginary part peaks [5].
Typically, the inductance remains constant below this rolloff
frequency, and decreases significantly in its vicinity and above
it due to increasing core loss. Therefore, the upper frequency
bound of the inductor is limited by the intrinsic loss peak. For
RF inductors, with SRF in the 100 MHz to 1 GHz range, which
is significantly higher than the material rolloff frequency for
typical Mn–Zn or Ni–Zn cores, it is inaccurate to assume that
the inductance remains at its low-frequency limit corresponding
to the initial permeability of the core. The permeability varies
around the rolloff frequency, and effectively, the equivalent cir-
cuit elements of the inductor become frequency-dependent.
Liu [6] described a method for extracting the self-capacitance
of the inductor by neglecting the influence of the core. Thus, it
is assumed that the capacitance with a ferrite core is unchanged
from its value for a winding with an air core (also see [16]).
For ferrite cores, this assumption is grossly violated because
both Mn–Zn and Ni–Zn cores have dielectric constants rang-
ing from 10 to 18 [7], [8]. Additionally, the high resistivity of
these materials has to be taken into account in evaluating the
shunt (leakage) resistance. Massarini and Kazimierczuk [9] de-
rived expressions for estimating the capacitance of multilayer
inductors by considering the approximate field distribution in
the vicinity of the core. The effects of the dielectric core on the
stray capacitance are not considered in their equations. Due to
the difficulty in accurately estimating the actual path of electric
flux or the surface area of the distributed capacitor, it is found that
the expressions given in [9] underestimate the capacitance by as
0018-9375/$26.00 © 2011 IEEE
924 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011
much as 40% when compared with an experimental method to
extract the stray capacitance from RF measurements [10].
The equivalent circuit parameters have been also obtained
from EM simulation methods, such as the quasi-static finite el-
ement [11] and the finite difference [12] methods. For toroidal
inductors, the intrinsic permeability in the frequency range from
10 kHz to 10 MHz has been computed using approximate so-
lution of steady-state-field equations to determine electric and
magnetic energies stored inside the core [13], [14]. Kazimier-
czuk et al. [15] evaluate integrals for core losses and the total
inductance of a solenoidal core inductor in terms of Bessel
functions derived from a solution to the Helmholtz equation in
cylindrical coordinates. Due to nature of the low-frequency ap-
proximations used, the solution in [15] is valid at frequencies up
to only a few MHz. In summary, the analyses in [11]–[15] are
specific to particular geometries of the inductor, involve time-
consuming numerical analysis, and are not applicable at RF due
to the quasi-static assumption.
Recently, there has been an interest to characterize RF induc-
tors using measurements on impedance and network analyzers.
Yu et al. [16] developed a method, based on network analyzer
measurements, to estimate the self-capacitance of inductors by
measuring the change in resonant frequency when a known (ex-
ternal) capacitor is shunted with the coil. The inductance and re-
sistance are determined from the capacitance and the frequency
shift. The inductance is assumed to vary slowly with frequency,
a conjecture strictly valid for frequencies below the rolloff fre-
quency of the core [10]. Due to the need to have closely spaced
external capacitance values for determining the frequency de-
pendence of the inductance, the experimental method in [16]
also suffers from limited frequency resolution. However, the re-
sults appear to be vastly improved over previous attempts [9] to
measure the stray capacitance. Using the well-known transmis-
sion line analysis of scattering parameters measured on coaxial
samples with a vector network analyzer, Shenhui and Quanx-
ing [17] extract the complex RF permeability of ferrite cores up
to 1 GHz. The cavity resonances of the sample and effect of air
gaps corrupt their results at the higher frequencies. In this pa-
per, we present a closed-form measurement-based model, which
comprehensively characterizes the RF inductor in terms of its
equivalent circuit representation.
The author developed a methodology for extracting in closed
form the frequency-dependent equivalent circuit parameters of
surface mount device (SMD) inductors [18], based on measured
S-parameters of inductors mounted on calibrated test boards.
This method has been extended to the equivalent circuit charac-
terization of other packaged RF components for printed circuit
applications, such as low-loss interconnects and SMD capaci-
tors [19]. The nonideal behavior associated with board layout
effects, device packaging, and component parasitic effects, has
been considered in deriving these models, which are valid over
a wide frequency band. Although not as versatile as commer-
cial CAD packages in the sense that the measurement-based
models in [18] and [19] are component-specific, they are de-
rived without recourse to cumbersome optimization procedures
normally followed in RF circuit synthesis. Because these com-
ponent models are developed in-situ in the same configuration
Fig. 1. Measurement setup for HP 4291 Impedance and Material Analyzer
(left) with a test station (right top) for mounting four sample fixtures (right)
used to measure discrete components and magnetic materials [20].
as their actual board layout, they can be easily incorporated
in circuit design, as we have demonstrated successfully for
a coplanar waveguide filter [19]. In this paper, we apply the
modeling methodology developed in [18] and [19] to extract
in closed form, the frequency-dependent RLC equivalent circuit
parameters of rod and toroidal ferrite inductors, from impedance
measurements. Expressions for complex intrinsic and rod per-
meability of toroidal and slug-type ferrite cores, respectively,
are also derived.
In the sequel, we briefly describe the measurement proce-
dure in Section II, followed by formulation of the closed-form
equivalent circuit model in Section III. Simple expressions are
presented to calculate the effective permeability in complex
form for both rod and toroidal core inductors. Sample results
on the equivalent circuit parameters and the effective perme-
ability of inductors mounted on ferrite rods and toroidal cores
are presented in Section IV, and validated with well-established
low-frequency models or manufacturer’s specifications where
applicable. Important conclusions are summarized in Section V.
II. MEASUREMENT PROCEDURE
The measurement setup, shown schematically in Fig. 1,
entails direct measurement of the RF impedance on the HP
4291 Material and Impedance Analyzer (MIA) [20]. The MIA
provides accurate impedance measurements from 1 MHz to
1.8 GHz with a high dynamic range (from about a tenth of an
ohm to few hundred kiloohms). Separate test heads are em-
ployed for measuring high and low impedances. An axial lead
fixture is connected to the test head for the impedance mea-
surement of toroidal or solenoidal inductors. Likewise, one uses
with the test head either a magnetic material fixture or a di-
electric material fixture, for the measurement of permeability
or permittivity, respectively. All these fixtures, shown in the
photograph in Fig. 1 [20], are internally compensated for elec-
trical length and parasitic impedance variations by calibrating
against known impedance standards. The relative permeability
of toroidal cores is calculated internally by the instrument from
NAISHADHAM: CLOSED-FORM DESIGN FORMULAS FOR THE EQUIVALENT CIRCUIT CHARACTERIZATION OF FERRITE INDUCTORS 925
Fig. 2. Equivalent circuit of an inductor.
measurement of the impedance on a one-turn “coil” (the con-
ducting path for the current is provided by a shorted coaxial
fixture). For impedance measurements, we have used the HP
16194 Axial Lead Impedance Fixture with a high impedance
test head. For toroidal core characterization, HP 16454 Mag-
netic Material Fixture and the low impedance test head have
been used. DC biasing and dynamic magnetization curves are
not considered in this study. Due to a large leakage flux path,
the rod permeability of solenoidal cores cannot be measured on
the impedance analyzer.
In addition to the complex impedance as a function of fre-
quency, the resonant frequency of the inductor was measured by
looking for the maximum of the resistive part. The quality (Q)
factor at resonance was measured from the resonant impedance
and the 3 dB bandwidth. The equivalent circuit characterization
and evaluation of other physical parameters, such as effective
permeability of the core, are then accomplished from these three
measured values—impedance, resonant frequency, and Q.
III. EQUIVALENT CIRCUIT REPRESENTATION
A. Closed-Form Circuit Model
At RF, the inductor is represented by the lumped-parameter
equivalent circuit shown in Fig. 2, where R,L, and Care the
equivalent series resistance, inductance, and stray capacitance,
respectively. Ris mainly caused by the resistive (skin effect)
losses in the winding, and the magnetic losses in the ferrite core.
The latter are predominantly due to imaginary (or resistive) part
of the permeability, and the core resistivity. At RF, these core
losses dominate the winding losses. Other core losses, such as
eddy current loss, hysterisis loss, and dissipation due to residual
magnetism or magnetic domain polarization, are neglected in
comparison with the resistive core losses.
The lumped parasitic capacitance Cincludes the turn-to-turn
distributed capacitance, turn-to-core capacitance, and for mul-
tilayer coils, also the layer-to-layer capacitance. In fact, for a
long coil, the capacitance is distributed, but if the operating fre-
quency and the circuit size are such that only the dominant reso-
nant mode is excited, then the premise of an equivalent lumped
capacitance is justified. For the RF coils that we measured, this
requirement was fulfilled. Thus, the equivalent circuit derived
in this paper is valid only for the dominant resonant mode.
Transmission line concepts will have to be used to extract the
high-frequency distributed equivalent circuit components of fer-
rite inductors, which support multiple resonant modes. We used
TAB LE I
MODEL PARAMETERS AT RESONANCE COMPUTED USING THE RESONANT
FREQUENCIES FOR THE TWO CASES DESCRIBED
connecting leads less than half an inch long. Therefore, the lead
inductance and the parasitic lead capacitance were ignored.
Denoting the measured complex impedance as Z(ω)=
Zr(ω)+jZi(ω),we obtain from Fig. 2
Z(ω)= R+jωL
1−ω2LC +jωRC (1)
where ω=2πf is the radian frequency. The impedance Z(ω)
in (1) presents two (real) equations at each frequency to deter-
mine the three variables, L, R, and C. However, as explained
shortly, the capacitance C, assumed to be frequency invariant,
can be explicitly computed from the resonant characteristics,
thus leaving two degrees of freedom for the calculation of Land
Rfrom the two independent equations in (1). Z(ω)resonates at
the frequency ω0with a quality factor Q, both of which can be
obtained explicitly from the measured data. The resonant fre-
quency is determined from measured data as the frequency at
which Re(Z)is maximum, which is close to the frequency de-
fined by the circuit theory criterion Im(Z)=0(see Table I and
relevant discussion) and Qis computed as Q=ω0/Δω, where
Δωis the measured 3 dB bandwidth of the impedance. Using
circuit theory, the resonant frequency and the quality factor for
Z(ω)may also be deduced from Fig. 2 as
ω0=1
√L0CQ2
1+Q21/2
(2)
Q=ω0L0
R0
(3)
where L0and R0denote inductance and resistance, respectively,
at the resonant frequency. The impedance Z(ω)in (1) may then
be rewritten in terms of resonant frequency as follows:
Z(ω)= R+jωL2C(ω2
0−ω2)
(1 −ω2LC)2+(ωRC)2.(4)
It is emphasized that at RF, both resistance Rand inductance L
are, in general, frequency-dependent, because the permeability
and resistivity of the ferrite-core material vary significantly with
frequency around resonance. The ferrite permittivity, however,
is not a strong function of frequency. Hence, we assume that the
stray capacitance C, which is affected only by the permittivity
of the ferrite core, remains constant with frequency.
At the resonant frequency ω0,let Z0≡Z(ω0)(a real num-
ber) be the impedance. Using (2) and (3) in (4), with ω=ω0
926 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011
substituted, we obtain at resonance
R0=Z0
1+Q2(5)
L0=QR0
ω0
.(6)
The self-capacitance, assumed to be constant about the first
resonant frequency, is then calculated using (2) and (6):
CΔ
=C(ω0)= 1
ω2
0L0
Q2
1+Q2.(7)
Therefore, knowledge of the measured resonant frequency,
quality factor, and the impedance at resonance, facilitates unique
determination of one of the three equivalent circuit parameters
for the RF inductor, namely, the frequency-invariant series ca-
pacitance C. After some algebra, the remaining two variables,
the frequency-dependent inductance and resistance, can also be
obtained in closed form in terms of the measured impedance
(see Appendix)
ωL(ω)= Zi(ω)+ωC |Z(ω)|2
[1 + ωCZi(ω)]2+[ωCZr(ω)]2(8)
R(ω)= Zr(ω)
[1 + ωCZi(ω)]2+[ωCZr(ω)]2.(9)
To the best of our knowledge, (7)–(9) are the most accurate ex-
pressions available to calculate the equivalent circuit parameters
of an RF inductor. These simple equations are valid at frequen-
cies beyond the SRF, and correctly predict the RF equivalent
circuit parameters for any inductor that can be characterized by
the equivalent circuit in Fig. 2. As shown in Section IV, the
model produces less than 1% error in comparison with the mea-
sured impedance for both toroidal and solenoidal inductors. A
similar agreement has been observed in [19] for SMD inductors.
As a CAD tool, this equivalent circuit model is expected to be
invaluable to RF engineers for the design and characterization
of EMI suppression components, as it simplifies the analysis
considerably without compromising accuracy.
A discussion on the capacitance calculation is in order.
Previous investigations [4], [6], [16] have estimated the self-
capacitance to be inversely proportional to the low-frequency L
instead of the inductance at resonance. However, the inductance
calculated from (8) for RF inductors yields a value at resonance
that is larger than its low-frequency value. Hence, it appears that
the previous studies overestimate the capacitance in compari-
son with the analytical formula (7), which uses the inductance
at resonance instead of the quasi-static value.
B. Calculation of Effective Permeability
The rod permeability of a slug, defined as the relative perme-
ability of a hypothetical magnetic circuit, which provides the
same reluctance as the magnetic core [5], is difficult to mea-
sure accurately because of the large magnetic flux leakage path
for cylindrical cores. Next, we will describe an approximate
method to deduce the effective permeability of slugs as well as
toroids, which yields consistent results and corroborates well
with available manufacturer’s specifications. As demonstrated
by the results in Section IV, this method provides the most ac-
curacy for RF inductors on toroidal cores, in which the intrinsic
permeability can be measured accurately because of a closed
magnetic flux path confined to the core.
In order to estimate the effective permeability of open cores
(such as ferrite slugs), which is much lower than the intrinsic
permeability for a toriodal core of the same material, we first
measure the impedance of a ferrite-core coil and calculate the
inductance and resistance from (8) and (9), respectively. Next,
we measure the impedance of an identical air-core coil (obtained
by removing the ferrite core from the coil) and calculate the in-
ductance Laand the resistance Ra, as described shortly. The
air-core coils have Q’s and resonant frequencies much higher
than those of the corresponding ferrite-core inductors. For ex-
ample, a 5-mm diameter, 20-mm long slug-type ferrite inductor
of 12 turns has a resonant frequency of 137 MHz and a Qof
10, whereas the corresponding coreless inductor resonates at
450 MHz and has a Qof 779. This means that the capacitance
of the air-core coil can be neglected, and the coil can be repre-
sented accurately by a series RL equivalent circuit. Therefore, if
we measure the complex impedance Zair =Rair +jXair of the
air-core coil, we can approximate Ra≈Rair and La≈Xair/ω
within the desired frequency range. The effective relative per-
meability μe=μ
e−jμ
eis then calculated using
μ
e(ω)=L(ω)
La
(10)
μ
e(ω)=R(ω)−Ra
ωLa
(11)
where the inductance Land the resistance Rfor the ferrite-
core inductor are computed using (8) and (9), respectively. The
permeability model in (10) and (11) assumes that the coil wind-
ing resistance is unchanged by the presence of the ferrite core,
i.e., if we write the resistance of the ferrite-core inductor as
R=Rc+Rw,where Rcaccounts for core losses and Rwfor
winding losses, then (11) assumes that the winding resistance
is only slightly perturbed from the value, Rw≈Ra.This is
true only if the current distribution in the coil is unchanged by
the additional magnetic flux introduced by the ferrite core. The
winding losses at high frequencies are dominated by the skin
effect, which depends strongly on the current distribution. An
investigation of the nonuniformity in current distribution intro-
duced by the ferrite core is needed in order to determine the
upper frequency limit of this assumption. The validity of this
assumption in (11) will be qualitatively established in Section
IV using measured data.
In the case of toroidal inductors, we have corroborated the
measured air-core inductance with the well-known static for-
mula available in standard references (cf., [2]). We have also
observed good agreement (within 10%) between the permeabil-
ity computed for a toroidal ferrite core using (10) and (11), and
the measured intrinsic permeability (see Fig. 6).
NAISHADHAM: CLOSED-FORM DESIGN FORMULAS FOR THE EQUIVALENT CIRCUIT CHARACTERIZATION OF FERRITE INDUCTORS 927
Fig. 3. Measured complex relative permeability of 4B1 toroidal core. (a) Real
part μ
r. (b) Imaginary part μ
r. Permeability measurements conducted using
HP 4291 Material Analyzer and HP 16454 Magnetic Material Fixture.
IV. MEASURED RESULTS
We have measured the impedance of toroidal and solenoidal
inductors, and calculated their equivalent circuit parameters, as
well as the effective permeability of the core, using the expres-
sions derived in the previous section. In this section, we present
the results on the following inductors:
1) A 6-turn coil (AWG 20.5) wound on a ferrite toroid
(Philips 4B1 nickel-zinc) of outer diameter 14 mm, inner
diameter 9 mm, and height 5 mm. This will be referred to
as Inductor 1.
2) A 12-turn coil (AWG 16) wound on a ferrite slug (Philips
4B1 nickel-zinc) of diameter 5 mm and length 20 mm.
This will be referred to as Inductor 2.
3) An 11-turn coil (AWG 18) wound on a ferrite slug (Fair-
Rite 43 nickel-zinc) of diameter 5 mm and length 15 mm.
This will be referred to as Inductor 3.
A. Toroidal Inductors
Fig. 3 displays the intrinsic complex relative permeability of
the toroidal 4B1 ferrite core from 1 MHz to 1.8 GHz, measured
on the HP 4291A Material Analyzer with the HP 16454 Mag-
netic Material Coaxial Fixture. The permeability of a rod core
cannot be measured directly because of the large leakage path
of the magnetic flux. Later in this section, we present results on
the effective rod permeability calculated from impedance mea-
surements for helical coils wound on slugs. The permeability at
1 MHz (see Fig. 3) is about 229 and relates to the low-frequency
initial permeability of 250 specified in the 1998 Philips Cata-
log (No. MA01) for Soft Ferrites [21]. The real part increases
to about 265 at 9 MHz, and then falls steadily with increasing
frequency, with a value of about 32 at 100 MHz. The rolloff
frequency corresponding to the magnetic loss peak occurs at 25
MHz, where the imaginary part is maximum. The traditional
low-frequency models of ferrite inductors (cf., [6]) are valid
below this frequency. An approximate value of saturation flux
density Bsat in the core can be calculated using Snoek’s law
given by 4πγBsat =μiωres, where μiis the initial relative per-
meability, γis the gyromagnetic ratio (a constant ∼1.76 ×1011
Fig. 4. Measured impedance of the 4B1 toroidal inductor.
C/kg in MKS units), and ωres =2πfres denotes the radian fre-
quency at the loss peak [5]. Since ferrite cores used in low-power
applications (such as cable television) are concerned with mag-
netic parameters below this frequency, rarely does the ferrite
manufacturer publish data for complex permeability above this
frequency. However, higher frequency data are essential when
specifying ferrite cores used in the suppression of RFI, and core
saturation effects take on paramount importance if high currents
are generated by the noise source (such as permanent magnet
DC motors in automotive applications).
Fig. 4 depicts the measured impedance of (toroidal) Inductor
1 over a frequency range from 1 to 500 MHz. The resonant
frequency (where the reactance is almost zero) is measured
as 102.16 MHz and the Qat resonance as 0.4352. The peak
resistance is about 906 Ω. It is clear from Fig. 4 that the reactance
is predominantly inductive below the resonant frequency and
capacitive above that frequency.
The equivalent series inductance and resistance of the toroidal
inductor, computed from the measured impedance using (8) and
(9), respectively, are plotted in Fig. 5 as functions of frequency.
For comparison, two curves are plotted: 1) equivalent circuit
parameters computed using the resonant frequency definition
corresponding to maximum Zr(ω), the real part of the measured
impedance; and 2) those computed using the resonant frequency
at which Zi(ω)0, where Zi(ω)is the imaginary part of the
measured impedance. For convenience, we refer to the former
as Case B, and the latter as Case C. We infer from Fig. 4 that
Case B gives a resonant frequency of 115.77 MHz, while Case
C yields 102.16 MHz, a discrepancy of 13%. The discrepancy is
due to the low Qof the inductor, and not an error in the measured
result. For high Qcircuits, these two definitions of the resonant
frequency yield the same result. Despite this discrepancy in
resonant frequencies, as summarized in Table I, the deduced
model parameters at resonance for these two cases, such as
equivalent inductance L0from (6) and capacitance C0from (7),
are all very close, suggesting that Case B definition of resonant
frequency is acceptable, and provides an accurate model. This
validation is very useful, because in practice, it is easier to
measure the peak of Zr(ω)than the zero or minimum of Zi(ω).
Henceforth, we will employ Case B definition.
The low-frequency inductance in Fig. 5 is 3.6 μHat1MHz
and decreases to 0.5 μH at coil resonance. This is a large swing
928 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011
Fig. 5. Computed (a) resistance and (b) inductance of the 4B1 toroidal
inductor.
in contrast to helical coils, which exhibit much smaller change
in inductance at resonance from its low-frequency value (see
Fig. 9). The reason is perhaps the complete trapping of the mag-
netic flux inside the toroidal core in contrast to a large leakage
flux outside the rod cores. This conjecture is supported by an or-
der of two magnitude reduction in the effective permeability of
the rod core in comparison with the material’s intrinsic perme-
ability, as will be demonstrated shortly. Equivalent capacitance
of the toroidal inductor, computed from (7) is 0.69 pF over the
entire band.
Fig. 6 displays the effective (in this case, intrinsic) perme-
ability of the 4B1 toroidal core, computed from the extracted
equivalent resistance and inductance of Inductor 1, as described
in Section III. The nominal low-frequency inductance Laof the
corresponding air core is about 0.2 μH. The winding resistance
Raremained less than 2.5 Ωbetween 10 and 500 MHz, which is
much smaller than the peak resistance of 906 Ωfor the ferrite in-
ductor. In order to validate the computed permeability in Fig. 6,
Fig. 6. Computed relative permeability of the six-turn toroidal inductor com-
pared with measured intrinsic permeability of a 14/9/5 4B1 ferrite toroid.
(a) Inductive permeability. (b) Resistive permeability.
measured values of the permeability are replotted from Fig. 3 at
a few frequencies. Excellent corroboration is observed between
model and measurement for the inductive permeability, whereas
about 12% discrepancy occurs in the resistive permeability at
frequencies near the rolloff frequency. This may be due to ne-
glecting the effect of ferrite core on the winding resistance (see
(11) and the discussion thereof).
For further verification of permeability calculations, we plot
in Fig. 6(a) the permeability obtained using the series induc-
tance calculated from measured impedance as L=Zi(ω)/ω,
the number of turns, and the geometric core constant C1=
ii/Ai,where iis the effective path length and Aiis the
area of cross section for a uniform (ith) segment of the toroidal
core. This quasi-static model neglects frequency-dependent ef-
fects. The core constant is calculated as C1=2.84 mm−1from
the toroid’s physical dimensions according to IEC Standards
Document 205, as outlined in [21]. The formula for the relative
NAISHADHAM: CLOSED-FORM DESIGN FORMULAS FOR THE EQUIVALENT CIRCUIT CHARACTERIZATION OF FERRITE INDUCTORS 929
Fig. 7. Measured impedance of the 4B1 solenoidal inductor (Inductor 2).
Fig. 8. Measured impedance of the Fair-Rite 43 (ATR) solenoidal inductor
(Inductor 3).
permeability is given by
μ
e(ω)= L×C1
1.257N2(12)
where Nis the number of turns, C1isinmm
−1, and Lis in
nanohenry. This inductive permeability, calculated using the
measured series inductance for L, is plotted in Fig. 6(a). It is
observed that the empirical formula (12) corroborates well with
measurements up to the rolloff frequency occurring at 25 MHz
(less than 5% error). At higher frequencies, this formula devi-
ates significantly from measurements, because it is derived from
a static magnetic flux distribution, which does not consider core
saturation, core losses, and frequency-dependent magnetic do-
main effects that become important above the rolloff frequency.
The accuracy of static formulas thus becomes questionable at the
higher frequencies, and the measured results presented in Fig. 3
should then be used for the inductive permeability of toroidal
cores. Alternatively, the frequency-dependent permeability ex-
tracted from the circuit model in (10) and (11), and plotted in
Fig. 6, may also be used. This conclusion emphasizes the ne-
cessity for engineers using ferrite cores for RFI suppression to
experimentally characterize the intrinsic permeability of core
materials at RF. The core manufacturer does not supply this
information, as it typically involves expensive RF instrumen-
tation. Instead, the manufacturer specifies the low-frequency
permeability computed in (12).
Fig. 9. Computed (a) resistance and (b) inductance of the 4B1 and ATR
soleniodal inductors.
B. Solenoidal (Helical) Inductors
Fig. 7 depicts the measured impedance of Philips 4B1 helical
inductor (no. 2) over a frequency range from 10 to 500 MHz. The
resonant frequency and the Qat resonance have been measured
as 137.4 MHz and 10.5, respectively, and the peak resistance is
about 12 kΩ. The inductor is designed for a nominal inductance
of 2 μH. It is again evident that the reactance is predominantly
inductive below the resonant frequency and capacitive above
that frequency. We plot in Fig. 8 the measured impedance of the
11-turn ATR inductor (no. 3) that employs Fair-Rite 43 ferrite
material, and is designed for a nominal inductance of 1.4 μH.
The resonant frequency is measured as 150.9 MHz and the Q
at resonance as 10. The peak resistance is about 8.2 kΩ.Next,
we evaluate the equivalent circuit parameters for both of these
inductors, computed from the measured impedance as described
in Section III.
The equivalent series inductance and resistance, obtained
from (8) and (9), respectively, are depicted in Fig. 9 as functions
930 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011
of frequency. In contrast to a toroidal inductor, resistance of the
helical inductor increases monotonically with frequency. For In-
ductor 2, the resistance reaches 250 Ωat the resonant frequency
of 137.4 MHz, and a peak value of 7.3 kΩat 387.3 MHz. The
small size of the solenoidal core causes the resistance peak at
the latter frequency, which appears to be pertinent to loss peak
in the magnetic core subject to leakage flux. Inductor 3 exhibits
a resistance of 210 Ωat the resonant frequency of 150.9 MHz,
and a peak value of 6 kΩat 370.4 MHz corresponding to loss
peak in permeability. The inductance stays relatively constant
well beyond the first resonance (from 10 to 300 MHz), with a
value between 1.98 and 2.15 μH for Inductor 2, and between
1.35 and 1.55 μH for Inductor 3. These values compare very
well with the nominal value of 2.0 and 1.4 μH, respectively,
for Inductors 2 and 3, measured at 20 kHz on a low-frequency
LCR bridge. Past the frequency at which resistance peaks, in-
ductance falls rapidly and becomes negative around 380 MHz,
causing the closed-form model in (8) to become invalid. There-
fore, inductance in Fig. 9 is not plotted beyond 400 MHz. The
frequency-independent stray capacitance, calculated from (7), is
0.66 and 0.8 pF, respectively, for Inductors 2 and 3. We may then
conclude that the equivalent circuit behavior with frequency is
consistent between these two inductors. It is worth mentioning
that the nominal capacitance of 0.8 pF for Inductor 3 is more
reasonable than the 4.188 pF that has been estimated using the
experimental method in [11]. In this paper [11], the capacitance
appears to be grossly overestimated for reasons cited following
(9) at the end of Section III-A.
Fig. 10 shows the effective permeability of the Philips 4B1
ferrite rod core (Inductor 2) and the Fair-Rite 43 ferrite rod
core (Inductor 3), computed using the equivalent series resis-
tance and inductance from Fig. 9, as described in Section III.
The striking similarity between the frequency dependence of
the toroidal core (intrinsic) permeability in Fig. 6 and the ef-
fective rod permeability in Fig. 10 is noted. We observe that
the rolloff around 400 MHz in the real part of permeability [see
Fig. 10(a)] is associated with the corresponding loss peak in
imaginary part of the permeability [see Fig. 10(b)]. This indi-
cates that the permeability rolloff in Fig. 10 indeed corresponds
to magnetic material behavior near material resonance, akin to
the permeability rolloff in Fig. 6. This important observation
needs further research for validation. The low-frequency induc-
tive rod permeability (at 10 MHz) is about 9.5 for the 4B1 core,
which is two orders of magnitude smaller than the intrinsic per-
meability (∼230) from Fig. 6, and can be justified physically
by the large leakage magnetic flux of the rod core. The induc-
tive permeability varies only slightly, between 8.5 and 9.5 for
Inductor 2, or 6 and 7 for Inductor 3, over a wide frequency
range spanning 10 to 300 MHz. This shows that our model is
very stable in spite of the large leakage flux and its effect on
inductance. The resistive permeability increases with frequency
till the material loss peaks around 400 MHz for both inductors,
and then falls to small values, akin to the resistive permeability
in Fig. 6 for the toroidal core. Over a span of 300 MHz, resistive
permeability also varies slightly, between 1 and 3.4 for Inductor
1, or between 0.2 and 2.5 for Inductor 2.
Fig. 10. Computed permeability of the 4B1 and ATR solenoidal inductors:
(a) inductive permeability and (b) resistive permeability.
C. Model Error
Since the RF equivalent circuit model for the inductor is
based on closed-form derivation of frequency-dependent cir-
cuit parameters from measured data, the error between model
and measurement is expected to be very small. This should
be contrasted with model parameters determined using numer-
ical optimization, in which case, the error is small only in se-
lective regions of validity (e.g., just around local minima of
the penalty function). We calculate the relative model error as
|(y−ˆy)/y|×100, where yis real (imaginary) part of the mea-
sured impedance, and ˆyis real (imaginary) part of the model
impedance. The latter is calculated by substituting in (1), the
equivalent circuit parameters determined from the measured
data using (7)–(9). As an example, Fig. 11 depicts the relative
model error for the 4B1 toroidal inductor. The maximum error is
less than 1% for both real and imaginary parts of the impedance,
thereby demonstrating accuracy of the equivalent circuit
NAISHADHAM: CLOSED-FORM DESIGN FORMULAS FOR THE EQUIVALENT CIRCUIT CHARACTERIZATION OF FERRITE INDUCTORS 931
Fig. 11. Percentage relative error between measurements and the equivalent
circuit model for the real and imaginary parts of the impedance of the 4B1
toroidal inductor. The model is calculated from (1) by substituting the equivalent
circuit parameters from (7)–(9). The model is in excellent agreement with the
measured data in Fig. 4, with maximum relative error less than 1%.
parameters, and validating the assumptions used in the model.
Similar corroboration has been observed for other inductors.
V. CONCLUSION
We have presented a rigorous experimental method to fully
characterize the RF behavior of ferrite-core inductors over a
wide frequency band, by considering all the parasitic effects,
such as flux leakage, stray capacitance, core losses, etc. It is be-
lieved that the proposed characterization method will be useful
to engineers designing RF noise suppression circuits in auto-
motive and wireless communications applications, where max-
imum noise suppression is desired in the frequency range from
100 MHz to 1 GHz, and no single reliable technique exists
to characterize inductors at these frequencies. The equivalent
circuit parameters of the inductor, as well as effective per-
meability of the core material, have been extracted in closed
form from measurements of the RF impedance, performed us-
ing an impedance analyzer. Sample results measured on several
ferrite inductors, both toroidal and solenoidal, yield consistent
data, which corroborate well with manufacturer’s specifications
where available, and with independent permeability measure-
ments on the HP 4291 Impedance/Material Analyzer for the
toroidal cores. The equivalent circuit model described in this
paper provides rigorously derived formulas to aid the design
of ferrite inductors. The proposed model is expected to shorten
the length of design cycles considerably by mitigating repetitive
measurements to characterize a wide array of inductors.
APPENDIX
We now derive the equivalent circuit model parameters in (8)
and (9) for the frequency-dependent inductance and resistance,
respectively. To derive the inductance L(ω)in (8), we begin
with the admittance in complex form, given by reciprocal of
Z(ω)in (1)
Y=Yr+jYi=R
R2+ω2L2+jω C0−L
R2+ω2L2.
(13)
It is emphasized that Rand Lare explicit functions of fre-
quency; the subscript on Cin (13) simply indicates that the
capacitance C0is constant with frequency, as given by (7). R
may be expressed in terms of Lusing the real part in (13)
R=1±1−(2YrωL)2
2Yr
,2YrωL < 1.(14)
Note that Ris positive for both signs before the radical in (14).
It is not necessary to choose one sign or the other, as clarified
in (17). A squaring operation removes the sign ambiguity. The
imaginary part in (13) may be written as follows:
Yi=ωC0−ωLYr
R.(15)
Substituting for Rfrom (14) in (15) yields an equation for L
of the form
X±X2−1=A, where A=Yr
ωC0−Yi
and X=1
2ωLYr
.
(16)
Transposing Xto the right side of (16), squaring both sides,
and solving for Xin terms of A, we obtain
X=1+A2
2A(17)
irrespective of the sign in front of the radical. Back substituting
for Xand Afrom (16) and rearranging terms results in the
sought-after expression for L(ω)
ωL(ω)= 1
ωC0+((|Y|2−ωC0Yi)/ωC0−Yi).(18)
Realizing that |Y|=1/|Z|,Y
r=Zr/|Z|2,and Yi=−Zi/
|Z|2, (18) after simplification yields (8).
Substitution of ωL from (18) into (15) results in the expres-
sion for the resistance
R(ω)= Yr
(ωC0)2−2ωC0Yi+|Y|2(19)
Once again, writing Yr,Y
i,and |Y|in terms of Zr,Z
i,and
|Z|, respectively, we obtain R(ω)in (9).
REFERENCES
[1] C. R. Paul, Introduction to Electromagnetic Compatibility, 2nd ed. New
York: John Wiley, 2006.
[2] H. W. Ott, Noise Reduction Techniques in Electronic Systems, 2nd ed.
New York: John Wiley, 1988.
[3] R.W.P.King,Electromagnetic Engineering, Vol. I: Fundamentals.New
York: McGraw-Hill, 1945, p. 463.
[4] M. Bartoli, N. Noferi, A. Reatti, and M. K. Kazimierczuk, “Modeling
winding losses in high-frequency power inductors,” J. Circuits, Syst.,
Comput., vol. 5, no. 4, pp. 607–626, 1995.
[5] E. C. Snelling, Soft Ferrites, Properties and Applications, 2nd ed.
Boston, MA: Butterworth, 1988.
[6] Y. Liu, “Power system equipment modeling based on wide frequency
external impedance measurements,” Ph.D. dissertation, The Ohio State
Univ., Columbus, OH, 1989.
[7] Soft Ferrite Components and Accessories, MMG Product Catalog, Issue
1A, 1995.
[8] Fair-Rite Soft Ferrites Catalog, 13th ed., Fair-Rite Products Corp., Wal-
lkill, NY, 1996.
[9] A. Massarini and M. K. Kazimierczuk, “Self-capacitance of inductors,”
IEEE Trans. Power Electron., vol. 12, no. 4, pp. 671–676, Jul. 1997.
932 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011
[10] K. Naishadham, “A rigorous experimental characterization of ferrite
inductors for RF noise suppression,” IEEE Radio Wireless Conf.,
Denver, CO, pp. 271–274, Aug. 1999.
[11] Q. Yu and T. W. Holmes, “A study on stray capacitance modeling of
inductors by using the finite element method,” IEEE Trans. Electromagn.
Compat., vol. 43, no. 1, pp. 88–93, Feb. 2001.
[12] R. F. Huang, D. M. Zhang, and K.-J. Tseng, “An efficient finite-difference-
based Newton_Raphson method to determine intrinsic complex perme-
abilities and permittivities for Mn–Zn ferrites,” IEEE Trans. Magn.,
vol. 42, no. 6, pp. 1655–1660, Jun. 2006.
[13] D. Zhang and C. F. Foo, “A practical method to determine intrinsic com-
plex permeabilities and permittivities for Mn–Zn ferrites,” IEEE Trans.
Magn., vol. 41, no. 4, pp. 1226–1232, Apr. 2005.
[14] R. F. Huang, D. M. Zhang, and K. -J. Tseng, “Determination of dimension-
independent magnetic and dielectric properties for Mn–Zn ferrite cores
and its EMI applications,” IEEE Trans. Electromagn. Compat., vol. 50,
no. 3, pp. 597–602, Aug. 2008.
[15] M. K. Kazimierczuk, G. Sancinento, G. Grandi, U. Reggiani, and A.
Massarini, “High-frequency small-signal model of ferrite core inductors,”
IEEE Trans. Magn., vol. 35, no. 5, pp. 4185–4191, Sep. 1999.
[16] Q. Yu, T. W.Holmes, and K. Naishadham, “RF equivalent circuit modeling
of ferrite core inductors and characterization of core materials,” IEEE
Trans. Electromagn. Compat., vol. 44, no. 1, pp. 258–262, Feb. 2002.
[17] J. Shenhui and J. Quanxing, “An alternative method to determine the
initial permeability of ferrite core using network analyzer,” IEEE Trans.
Electromagn. Compat., vol. 47, no. 3, pp. 651–657, Aug. 2005.
[18] K. Naishadham, “Extrinsic equivalent circuit modeling of SMD inductors
for printed circuit applications,” IEEE Trans. Electromagn. Compat.,
vol. 43, no. 4, pp. 557–565, Nov. 2001.
[19] K. Naishadham and T. Durak, “Measurement-based closed-form modeling
of surface-mounted RF components,” IEEE Trans. Microw. Theory Tech.,
vol. 50, no. 10, pp. 2276–2286, Oct. 2002.
[20] Measurement of Impedance using the HP 4291 Impedance Analyzer,HP
Product Note 4291-1, Hewlett-Packard, Palo Alto, CA, 1994.
[21] Philips Soft Ferrites Data Handbook, MA-01, Philips, Amsterdam, The
Netherlands, 1998.
Krishna Naishadham received the M.S. degree from
Syracuse University, Syracuse, NY, and the Ph.D. de-
gree from the University of Mississippi, Oxford, MS,
both in electrical engineering, in 1982 and 1987, re-
spectively.
He was engaged with the faculty of electrical
engineering for 15 years at the University of Ken-
tucky, Wright State University as Tenured Professor,
and Syracuse University as an Adjunct Professor. He
taught courses in electromagnetics, microwave engi-
neering and antennas, and performed sponsored and
unsponsored research on a variety of applied electromagnetics (EM) topics,
graduating three Ph.D. students and several M.S. students. In 2002, he joined
Lincoln Laboratory, Massachusetts Institute of Technology, Cambridge, MA
as a Research Scientist and contributed innovative asymptotic techniques and
spectral estimation methods for the electromagnetic signature analysis of large
objects containing sections with small features. In 2008, he joined Georgia In-
stitute of Technology, Atlanta, GA, where he holds a joint appointment with the
Georgia Tech Research Institute as a Principal Research Scientist, and the ECE
Department as a Research Professor. He is currently leading research projects on
novel multifunctional antenna design for aerial platforms, and carbon-nanotube-
based chemical sensors. He served as an Associate Editor for the Applied Com-
putational Electromagnetics Society Journal. He is currently an Associate Edi-
tor of the International Journal of Microwave Science and Technology.Hehas
authored or coauthored four book chapters and more than 150 papers in profes-
sional journals and conference proceedings on topics related to computational
EM, high-frequency asymptotic methods, antenna design, EMC, materials char-
acterization, and wave-oriented signal processing.
Dr. Naishadham is currently the Chair of the Joint IEEE AP/MTT Chapter at
Atlanta and serves on the Technical Program Committee for the International
Microwave Symposium.
